Escape rates and physical measures for the infinite horizon Lorentz gas with holes

Abstract

We study the statistical properties of the infinite horizon Lorentz gas after the introduction of small holes. Our basic approach is to prove the persistence of a spectral gap for the transfer operator associated with the billiard map in the presence of such holes. The new feature here is the interaction between the holes and the infinite horizon corridors, which causes previous approaches to fail. In order to overcome this difficulty, we redefine the Banach spaces on which we consider the action of the transfer operator. In this modified setting, we recover a complete set of results for the open system: Existence of a unified exponential rate of escape and limiting conditionally invariant measure for a large class of initial distributions, the convergence of the physical conditionally invariant measure to the smooth invariant measure for the billiard as the size of the hole tends to zero and the characterization of the escape rate via a notion of pressure on the survivor set.

Keywords:

• Open systems
• escape rates
• dispersing billiards
• transfer operator
• conditionally invariant measure
• survivor set

Acknowledgements

This research is partially supported by NSF grant DMS-1101572.

Notes

1. In the non-invertible case, we define .

2. Here by , we mean with p = 1.

3. Our treatment of stable curves here differs from that in [20]. In that abstract setting, stable curves are defined via graphs in charts of the given manifold. In the present more concrete setting, we dispense with charts and use the global (r, ϕ) coordinates.

4. Recall that a physical measure for T is an ergodic, invariant probability measure μ for which there exists a positive Lebesgue measure set B _μ, with μ(B _μ) = 1, such that for all x ∈ B _μ and all continuous functions ψ.

5. Indeed, [21] shows only the bound ^1/2 in the finite horizon case, but a quick calculation shows that an exponent of 1/3 is in fact needed in the infinite horizon case.